Download ON GENERALIZED w-CLOSED SETS IN w

International Journal of Pure and Applied Mathematics
Volume 110 No. 2 2016, 327-335
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
doi: 10.12732/ijpam.v110i2.7
AP
ijpam.eu
ON GENERALIZED w-CLOSED SETS IN w-SPACES
Won Keun Min1 § , Young Key Kim2
1 Department
of Mathematics
Kangwon National University
Chuncheon, 200-701, KOREA
2 Department of Mathematics
MyongJi University
Yongin, 449-728, KOREA
Abstract:
The purpose of this note is to introduce the notions of generalized w-closed set
and generalized w-open set in w-spaces. In fact, every w-open set is generalized w-open in
a given w-space. We study some basic properties of such the notions and the conditions of
W -continuous functions which preserve generalized w-closed sets or generalized w-open sets.
AMS Subject Classification: 54A05, 54B10, 54C10
Key Words: w-spaces, generalized w-open(w-closed), W (W ∗)-continuous, W ∗ -closed function, quasi-W ∗ -closed function.
1. Introduction
Siwiec [17] introduced the notions of weak neighborhoods and weak base in a
topological space. We introduced the weak neighborhood systems defined by
using the notion of weak neighborhoods in [13]. The weak neighborhood system induces a weak neighborhood space which is independent of neighborhood
spaces [4] and general topological spaces [2]. The notions of weak structure, wspace, W -continuity and W ∗ -continuity were investigated in [14]. In fact, the
set of all g-closed subsets [5] in a topological space is a kind of weak structure.
Received:
Revised:
Published:
June 31, 2016
September 14, 2016
November 4, 2016
§ Correspondence author
c 2016 Academic Publications, Ltd.
url: www.acadpubl.eu
328
W.K. Min, Y.K. Kim
The one purpose of our research is to introduce the notion of generalize
w-open sets (generalize w-closed sets)in w-spaces. Levine [5] introduced the
notion of g-closed subsets in topological spaces. In the same way, we introduce
the notion of generalized w-closed set (simply, gw-closed set) in weak spaces,
and investigate some basic properties of such notions.
2. Preliminaries
Definition 2.1 ([14]). Let X be a nonempty set. A subfamily wX of the
power set P (X) is called a weak structure on X if it satisfies the following:
(1) ∅ ∈ wX and X ∈ wX .
(2) For U1 , U2 ∈ wX , U1 ∩ U2 ∈ wX .
Then the pair (X, wX ) is called a w-space on X. Then V ∈ wX is called a
w-open set and the complement of a w-open set is a w-closed set.
The collection of all w-open sets (resp., w-closed sets) in a w-space X will be
denoted by W O(X) (resp., W C(X)). We set W (x) = {U ∈ W O(X) : x ∈ U }.
Let S be a subset of a topological space X. The closure (resp., interior)
of S will be denoted by clS (resp., intS). A subset S of X is called a preopen
set [11] (resp., α-open set [16], semi-open [6]) if S ⊂ int(cl(S)) (resp., S ⊂
int(cl(int(S))), S ⊂ cl(int(S))). The complement of a preopen set (resp., αopen set, semi-open) is called a preclosed set (resp., α-closed set, semi-closed).
The family of all preopen sets (resp., α-open sets, semi-open sets) in X will
be denoted by P O(X) (resp., α(X), SO(X)). We know the family α(X) is a
topology finer than the given topology on X.
A subset A of a topological space (X, τ ) is said to be:
(a) g-closed [5] if cl(A) ⊂ U whenever A ⊂ U and U is open in X;
(b) gp-closed [7] if pCl(A) ⊂ U whenever A ⊂ U and U is open in X;
(c) gs-closed [1, 3] if sCl(A) ⊂ U whenever A ⊂ U and U is open in X;
(d) gα-closed [9] if τ α Cl(A) ⊂ U whenever A ⊂ U and U is α-open in X
where τ α = α(X);
(e) gα∗ -closed [8] if τ α Cl(A) ⊂ int(U ) whenever A ⊂ U and U is α-open in
X;
(f) gα∗∗ -closed [8] if τ α Cl(A) ⊂ int(cl(U )) whenever A ⊂ U and U is α-open
in X;
(g) αg-closed [9] if τ α Cl(A) ⊂ U whenever A ⊂ U and U is open in X;
(h) α∗∗ g-closed [9] if τ α Cl(A) ⊂ int(cl(U )) whenever A ⊂ U and U is open
in X.
ON GENERALIZED w-CLOSED SETS IN w-SPACES
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Then the family τ , GO(X), gαO(X), gα∗ O(X), gα∗∗ O(X), αgO(X) and
are all weak structures on X. But P O(X), GP O(X) and SO(X)
are not weak structures on X. A subfamily mX of the power set P (X) of a
nonempty set X is called a minimal structure on X [10] if ∅ ∈ wX and X ∈ wX .
Thus clearly every weak structure is a minimal structure.
α∗∗ gO(X)
Definition 2.2 ([14]). Let (X, wX ) be a w-space. For a subset A of X,
the w-closure of A and the w-interior of A are defined as follows:
(1) wC(A) = ∩{F : A ⊆ F, X − F ∈ wX }.
(2) wI(A) = ∪{U : U ⊆ A, U ∈ wX }.
Theorem 2.3 ([14]). Let (X, wX ) be a w-space and A ⊆ X.
(1) x ∈ wI(A) if and only if there exists an element U ∈ W (x) such that
U ⊆ A.
(2) x ∈ wC(A) if and only if A ∩ V 6= ∅ for all V ∈ W (x).
(3) If A ⊂ B, then wI(A) ⊂ wI(B); wC(A) ⊂ wC(B).
(4) wC(X − A) = X − wI(A); wI(X − A) = X − wC(A).
(5) If A is w-closed (resp., w-open), then wC(A) = A (resp., wI(A) = A).
3. Main results
Definition 3.1. Let (X, wX ) be a w-space and A ⊆ X. Then A is called
a generalized w-closed set (simply, gw-closed set) if wC(A) ⊆ U , whenever
A ⊆ U and U is w-open.
Remark 3.2. (1) If the wX -structure is a topology, the generalized wclosed set is exactly a generalized closed set in sense of Levine in [5].
(2) Obviously, every w-closed set is generalized w-closed, but in general, the
converse is not true as the next example.
Example 3.3. Let X = {a, b, c, d} and wX = {∅, {a}, {b}, {c}, {a, d}, X}
be a w-structure in X. Now, consider A = {d}. Then wC(A) = A. So A is
gw-closed but not w-closed.
Lemma 3.4. Let (X, wX ) be a w-space and A, B ⊆ X. Then the following
things hold:
(1) wI(A) ∩ wI(B) = wI(A ∩ B).
(2) wC(A) ∪ wC(B) = wC(A ∪ B).
Theorem 3.5. Let (X, wX ) be a w-space. Then the union of two gwclosed sets is a gw-closed set.
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Proof. Let A and B be any two gw-closed sets. Let G be any w-open set
such that A ∪ B ⊆ G. Then A ⊆ G and B ⊆ G. Since A and B are gwclosed sets, wC(A) ⊆ G and wC(B) ⊆ G. Now, by Lemma 3.4, wC(A ∪ B) =
wC(A) ∪ wC(B) ⊆ G. Hence A ∪ B is gw-closed.
In general, the intersection of two gw-closed sets is not gw-closed:
Example 3.6. For X = {a, b, c, d}, let w = {∅, {a, c}, {a}, {b}, {c}, {a, d}, X}
be a w-structure in X. Now, consider A = {a, b, c} and B = {a, c, d}. Then
A and B are gw-closed. But A ∩ B = {a, c} is not gw-closed because {a, c} is
w-open and wC({a, c}) = {a, c, d}.
Theorem 3.7. Let (X, wX ) be a w-space. Then if A is a gw-closed set,
then wC(A) − A contains no non-empty w-closed set.
Proof. Suppose that there is a w-closed set F such that F ⊆ wC(A) − A.
Then A ⊆ X−F , and since X−F is w-open and A is gw-closed, wC(A) ⊆ X−F .
It implies that F ⊆ X − wC(A), and so F ⊆ wC(A)∩ (X − wC(A)) = ∅. Hence,
F = ∅.
In general, the converse in Theorem 3.7 is not true as shown in the next
example.
Example 3.8. Let X = {a, b, c, d} and wX = {∅, {a}, {b}, {c}, {a, d}, X}
be a w-structure in X. Consider A = {a}. Note wC(A) = {a, d} and wC(A) −
A = {a, d} − {a} = {d}. Since {d} is not w-closed, wC(A) − A contains no
non-empty w-closed set, but A is not w-closed.
Theorem 3.9. Let (X, wX ) be a w-space. Then if A is a gw-closed set
and A ⊆ B ⊆ wC(A), then B is gw-closed.
Proof. Let U be any w-open set such that B ⊆ U . Since A ⊆ B ⊆ wC(A)
and A is a gw-closed set, A ⊆ U and wC(B) ⊆ wC(A) ⊆ U . It implies that
wC(B) ⊆ U , and so B is gw-closed set.
Definition 3.10. Let (X, wX ) be a w-space and A ⊆ X. Then A is called
a generalized w-open set (simply, gw-open set) if X − A is gw-closed.
Theorem 3.11. Let (X, wX ) be a w-space and A ⊆ X. Then A is
generalized w-open if and only if F ⊆ wI(A) whenever F ⊆ A and F is wclosed.
Proof. From Definition 3.1, it is obvious.
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Theorem 3.12. Let (X, wX ) be a w-space. Then the intersection of two
gw-open sets is a gw-open set.
Proof. It is obvious from Theorem 3.5.
In general, the union of two gw-open sets is not gw-open (See Example 3.6).
Theorem 3.13. Let (X, wX ) be a w-space and A ⊆ X. Then if A is
gw-open, then U = X, whenever wI(A) ∪ (X − A) ⊆ U and U is w-open.
Proof. Let U be any w-open set and wI(A) ∪ (X − A) ⊆ U . Then X − U ⊆
(X − wI(A)) ∩ A = wC(X − A) ∩ A = wC(X − A) − (X − A). Since X − A
is gw-closed, by Theorem 3.7, the w-closed set X − U must be empty. Hence,
U = X.
Theorem 3.14. Let (X, wX ) be a w-space. Then if A is a gw-open set
and wI(A) ⊆ B ⊆ A, then B is gw-open.
Proof. It is similar to the proof of Theorem 3.9.
Theorem 3.15. Let (X, wX ) be a w-space. Then if A is a gw-closed set,
then wC(A) − A is gw-open.
Proof. Suppose that A is a gw-closed set. Then by Theorem 3.9, the empty
set is the only one w-closed subset of wC(A) − A. So, for the only w-closed
subset ∅ of wC(A) − A, ∅ ⊆ wC(A) − A and ∅ ⊆ wI(wC(A) − A). Hence,
wC(A) − A is gw-open.
Theorem 3.16. Let (X, wX ) be a w-space. Then if A is a gw-open set,
then wI(A) ∪ (X − A) is gw-closed.
Proof. Suppose that A is a gw-open set. Then by Theorem 3.13, the whole
set X is the only one w-open set containing wI(A) ∪ (X − A). So, wI(A) ∪
(X − A) ⊆ X and wC(wI(A) ∪ (X − A)) ⊆ X. Hence, wI(A) ∪ (X − A) is
gw-closed.
Definition 3.17. Let (X, wX ) be a w-space. For a subset A of X, gwclosure of A and gw-interior of A are defined as the following:
(1) gwC(A) = ∩{F : A ⊆ F, F is gw-closed}.
(2) gwI(A) = ∪{U : U ⊆ A, U is gw-open}.
Theorem 3.18. Let (X, wX ) be a w-space and A ⊆ X.
(1) If A is gw-open, then gwI(A) = A.
(2) If A is gw-closed, then gwC(A) = A.
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Proof. Obvious.
But the converses in the above theorem are not always true as shown in the
next example.
Example 3.19. In Example 3.6, let F = {a, c}. Since {a, b, c} and {a, c, d}
are gw-closed sets, gwC(F ) = {a, c}. But from the fact that {a, c} is w-open
and wC({a, c}) = {a, c, d}, F is not gw-closed. Similarly, we can show that the
converse of (2) in Theorem 3.18 is not true, in general.
Theorem 3.20. Let (X, wX ) be a w-space and A, B ⊆ X.
(1) If A ⊆ B, then gwI(A) ⊆ gwI(B) and gwC(A) ⊆ gwC(B).
(2) gwC(X − A) = X − gwI(A); gwI(X − A) = X − gwC(A).
Proof. Obvious.
Theorem 3.21. Let (X, wX ) be a w-space and A ⊂ X.
(1) x ∈ gwI(A) if and only if there exists a gw-open set U containing x
such that U ⊆ A.
(2) x ∈ gwC(A) if and only if A ∩ V 6= ∅ for all gw-open set V containing
x.
Proof. (1) It is obvious.
(2) For each x ∈ gwC(A), suppose that there exists a gw-open set V containing x such that A ∩ V = ∅. Then A ⊆ X − V , so gwC(A) ⊆ X − V and
x∈
/ gwC(A), which is a contradiction.
Conversely, suppose on the contrary that x ∈
/ wC(A). Then there exists a
gw-closed set F containing A such that x ∈
/ F . It implies that there exists a
gw-open set X − F containing x such that (X − F ) ∩ A = ∅. So, the proof is
completed.
Theorem 3.22. Let (X, wX ) be a w-space and A, B ⊂ X.
(1) ∅ = gwC(∅).
(2) A ⊆ gwC(A).
(3) gwC(A ∪ B) = gwC(A) ∪ gwC(B).
(4) gwC(gwC(A)) = gwC(A).
Proof. (1) and (2) are obvious.
(3) From Theorem 3.20, gwC(A ∪ B) ⊇ gwC(A) ∪ gwC(B). Now, we show
that gwC(A ∪ B) ⊆ gwC(A) ∪ gwC(B). For the proof, let x ∈
/ gwC(A) ∪
gwC(B). Then there exist gw-closed sets F1 and F2 such that x ∈
/ F1 and
A ⊆ F1 ; x ∈
/ F2 and B ⊆ F2 . So x ∈
/ F1 ∪ F2 and A ∪ B ⊆ F1 ∪ F2 . Since F1 ∪ F2
ON GENERALIZED w-CLOSED SETS IN w-SPACES
333
is gw-closed, we have that x ∈
/ gwC(A ∪ B). Consequently, gwC(A ∪ B) =
gwC(A) ∪ gwC(B).
(4) It is sufficient to show that gwC(gwC(A)) ⊆ gwC(A). For any gw-closed
set F satisfying A ⊆ F , gwC(A) ⊆ gwC(F ) = F . From the fact, {F : A ⊆
F, F is gw-closed} ⊆ {K : gwC(A) ⊆ K, K is gw-closed}. So gwC(gwC(A)) =
∩{K : gwC(A) ⊆ K, K is gw-closed} ⊆ ∩{F : A ⊆ F, F is gw-closed} =
gwC(A).
Theorem 3.23. Let (X, wX ) be a w-space and A, B ⊂ X.
(1) X = gwI(X).
(2) gwI(A) ⊆ A.
(3) gwI(A ∩ B) = gwI(A) ∩ gwI(B).
(4) gwI(gwI(A)) = gwI(A).
Proof. These are easily obtained by Theorem 3.20 and Theorem 3.22.
Finally, we have a topology induced by generalized w-open sets as the following:
Theorem 3.24. Let (X, wX ) be a w-space. Then the family τ ∗ = {U ⊆
X : U = gwI(U )} is a topology containing wX .
Let X and Y be w-spaces. A function f : (X, wX ) → (Y, wY ) is said to be
W ∗ -open [15] if for every w-open set G in X, f (G) is a w-open set in Y .
In the analogous way, we define the following notions:
Definition 3.25. Let X and Y be w-spaces. A function f : (X, wX ) →
(Y, wY ) is said to be
(1) W ∗ -closed if for every w-closed set F in X, f (F ) is a w-closed set in Y .
(2) quasi-W ∗ -closed if for A ⊆ X, wC(f (A)) ⊆ f (wC(A)).
Remark 3.26. There is no any relation between the notions of W ∗ -closed
function is quasi-W ∗ -closed function.
Example 3.27. Let X = {a, b, c} and w1 = {∅, {a}, {b}, X} be a wstructure in X.
(1) Let w2 = {∅, {b}, {c}, X} be a w-structure in X. Consider a function
f : (X, w1 ) → (X, w2 ) defined by f (a) = f (b) = a; f (c) = b. Then the function
is obviously W ∗ -closed. Now, note that for A = {c}, wC(A) = A in w1 and
f (wC(A)) = f (A) = {b}; wC(f (A)) = wC({b}) = {a, b} in w2 . So, f is not
quasi-W ∗ -closed.
(2) Let w3 = {∅, {a, b}, X} be a w-structure in X. Consider a function
f : (X, w3 ) → (X, w1 ) defined by f (a) = b; f (b) = a; f (c) = c. Note that in the
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W.K. Min, Y.K. Kim
w-space (X, w3 ), for nonempty set A, if A = {c}, then wC(A) = A; otherwise,
wC(A) = X. So, f is quasi-W ∗ -closed. But for a w-closed set A = {c} in
(X, w3 ), since f (A) = {c} is not w-closed in (X, w1 ), f is not W ∗ -closed.
Recall the notion of W -continuity: Let (X, wX ) and (Y, wY ) be two wspaces. Then f : X → Y is said to be W-continuous [14] if for x ∈ X and
V ∈ W (f (x)), there is U ∈ W (x) such that f (U ) ⊆ V .
Theorem 3.28 ([14]). Let f : (X, wX ) → (Y, wY ) be a function in two
w-spaces. Then the following statements are equivalent:
(1) f is W -continuous.
(2) f (wC(A)) ⊆ wC(f (A)) for A ⊆ X.
(3) wC(f −1 (B)) ⊆ f −1 (wC(B)) for B ⊆ Y .
(4) f −1 (wI(B)) ⊆ wI(f −1 (B)) for B ⊆ Y .
Lemma 3.29. Let f : X → Y be a function between w-spaces X and Y .
(1) If f is W ∗ -closed, then f (wC(A)) ⊆ wC(f (A)) for each A ⊆ X.
(2) f is W -continuous and quasi-W ∗ -closed if and only if wC(f (A)) =
f (wC(A)) for every subset A in X.
Proof. (1) Straightforward.
(2) From the notion of quasi W ∗ -closed function and property of W -continuity
in Theorem 3.28, directly the statement is obtained.
Theorem 3.30. Let f : (X, wX ) → (Y, wY ) be a function on w-spaces X
and Y . Then the following statements hold:
(1) If f is W -continuous and W ∗ -closed, then for every gw-open subset B
in Y , f −1 (B) is gw-open.
(2) If f is W ∗ -continuous and quasi-W ∗ -closed, for every gw-closed set A
in X, f (A) is gw-closed.
Proof. (1) Let B be any gw-open subset B in Y , and F be a w-closed set
in X such that F ⊆ f −1 (B). Now, we show that F ⊆ wI(f −1 (B)). Since f
is W ∗ -closed, f (F ) is w-closed. Moreover, since B is gw-open, f (F ) ⊆ wI(B).
From Theorem 3.28, it follows that F ⊆ f −1 (wI(B)) ⊆ wI(f −1 (B)). Hence,
f −1 (B) is gw-open.
(2) Let A be any gw-closed subset A in X, and U be a w-open set in Y
such that f (A) ⊆ U . Now, we show that wC(f (A)) ⊆ U . Since f is W ∗ continuous and A is gw-closed, f −1 (U ) is w-open and wC(A) ⊆ f −1 (U ). Since
f is quasi-W ∗ -closed, wC(f (A)) ⊆ f (wC(A)) ⊆ f f −1 (U ) ⊆ U , and hence,
f (A) is gw-closed.
ON GENERALIZED w-CLOSED SETS IN w-SPACES
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